Which of the following equations is dimensionally incorrect?

Question:

Which of the following equations is dimensionally incorrect?

Where $\mathrm{t}=$ time, $\mathrm{h}=$ height, $\mathrm{s}=$ surface tension, $\theta=$ angle,$\rho=$ density, a, $r=$ radius, $\mathrm{g}=$ acceleration due to gravity, $\mathrm{v}=$ volume, $\mathrm{p}=$ pressure, $\mathrm{W}=$ work done, $\Gamma=$ torque, $\in=$ permittivity, $\mathrm{E}=$ electric field, $\mathrm{J}=$ current density, $\mathrm{L}=$ length.

  1. $\mathrm{v}=\frac{\pi \mathrm{pa}^{4}}{8 \eta \mathrm{L}}$

  2. $\mathrm{h}=\frac{2 s \cos \theta}{\rho \mathrm{rg}}$

  3. $\mathrm{J}=\epsilon \frac{\partial \mathrm{E}}{\partial \mathrm{t}}$

  4. $\mathrm{W}=\Gamma \theta$


Correct Option: 1

Solution:

(i) $\frac{\pi \mathrm{pa}^{4}}{8 \eta \mathrm{L}}=\frac{\mathrm{dv}}{\mathrm{dt}}=$ Volumetric flow rate

(poiseuille's law)

(ii) $\mathrm{h} \rho \mathrm{g}=\frac{2 \mathrm{~s}}{\mathrm{r}} \cos \theta$

(iii) $\mathrm{RHS} \Rightarrow \varepsilon \times \frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{a}}{\mathrm{r}^{2}} \times \frac{1}{\varepsilon}=\frac{\mathrm{q}}{\mathrm{t}} \times \frac{1}{\mathrm{r}^{2}}$

$=\frac{\mathrm{I}}{\mathrm{L}^{2}}=\mathrm{IL}^{-2}$

LHS

$\mathrm{T}=\frac{\mathrm{I}}{\mathrm{A}}=\mathrm{IL}^{-2}$

(iv) $\mathrm{W}=\tau \theta$

Option (1)

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